can someone help me differentiate please x^2 * square root of (x+2)?

Question

the answer i cant seem to get to is x(5x-8)/2 square root(x-2)
and i wrote the problem incorrectly it is -2 not +2

Answer 1

f (x) = x² (x - 2)^(1/2)
f `(x) = (2x)(x - 2)^(1/2) + (1/2)(x - 2)^(-1/2)(x²)
f `(x) = (1/2)(x - 2)^(-1/2) [ 4x (x - 2) + x² ]
f `(x) = (1/2) [ 5x² - 8x ] / (x - 2)^(1/2)
f `(x) = [ (x) (5x - 8) ] / [ 2 (x - 2)^(1/2) ]

Answer 2

Use the product rule. Take the derivative of x^2 and multiply this by √(x+2). And add to this the product of x^2 and the derivative of √(x+2).

[ 2x * √(x+2) ] + [x^2 * (1/2)(x+2)^(-1/2)]

Now go simplify.

Answer 3

I'll rewrite the problem in a way that will make it easier to differentiate:
x^2 * (x+2)^(1/2)
Then use the multiplication rule (d/dx (f * g) = f * d/dx g + g * d/dx f)
derivative is:
x^2 * 1/2(x+2)^(-1/2) + 2x * (x+2)^(1/2)
= x^2 * 1/2 * 1/sqr rt(x+2) + 2x * sqr rt(x+2)